{"id":"54234b5c-2d17-45c2-8631-b35939b208be","arxiv_id":"1908.08948","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"A noncommutative polynomial with infinitely many factoring scalar fibers must be a univariate composition p(h), and this characterizes eigenlevel set inclusion and local quasiconvexity.","lead":"This paper proves a free algebra version of Bertini's theorem: if a noncommutative polynomial f minus a scalar factors for infinitely many scalars, then f must be a composition p(h) of a univariate polynomial with another noncommutative polynomial. It applies the theorem to matrix eigenvalue sets and to convexity of matrix positivity domains.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the central theorem's proof is sound given the cited centralizer results; only the applications' exposition is terse.","rationale":"After tracing the proof of Theorem 3.2, the argument is coherent: (iv) to (ii) is straightforward, (iii) to (iv) is Bergman, and (i) to (iii) is the substantive direction. In that direction, the pigeonhole argument with Proposition 2.1 and Lemma 3.1 is sound; the determinant normalization uses Lemma 2.4 and works because the relevant generic determinants are nonconstant; the stable-association step follows from Cohn's criteria with the appropriate coprimality conditions. The only non-constructive step is the extension of the centralizer to the universal skew field, but this is explicitly attributed to published sources. The proof of Lemma 2.2 is elementary and correct. The descent at the end of (i) to (iii), though terse, is valid because a nonzero subspace defined over the base field contains base points if it contains algebraic-closure points. I therefore believe the central claim is sound. The reader's conditional verdict stems from an expository gap in Theorem 5.4, not from a defect in the main theorem; I would not change the verdict.","tokens_in":11178,"tokens_out":36009,"duration_ms":343131,"concrete_test":"Verify the exact hypotheses of [Coh06, Theorem 7.9.8]: check that it states the centralizer of a non-composite element of a free algebra in the universal field of fractions equals the fraction field of the centralizer in the free algebra; if the theorem instead requires additional hypotheses, apply it to Lemma 3.1 with the stated assumptions and confirm they are met. This would settle whether Lemma 3.1's conclusion is fully supported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I reviewed the proof of Theorem 3.2 and its key Lemma 3.1. The most delicate step is the assertion that the centralizer of a non-composite f in the universal skew field F(<x>) equals F(f), cited to [Coh06, Theorem 7.9.8 and Proposition 3.2.9]. If this cited theorem failed, Lemma 3.1 would collapse. However, the step is a standard consequence of Bergman's centralizer theorem: non-composite f has centralizer F[f] in F<x>, and the universal field of fractions extends this to F(f); the determinant argument in Lemma 3.1 supplies the needed scalar normalization and the stable-association argument supplies the proportionality. I found no internal inconsistency or missing computation in the proof. The separate terse passage in Theorem 5.4, where h is asserted to be symmetric due to uniqueness up to scalar multiple, is an exposition gap rather than a mathematical error. Therefore I do not identify a load-bearing concern with the paper's central claim.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves a free-algebra analogue of Bertini's irreducibility theorem (Theorem 3.2): for a nonconstant noncommutative polynomial f, if f - λ factors for infinitely many scalars λ, then f = p ∘ h for some noncommutative h and univariate p of degree greater than 1; equivalently, a non-composite f has irreducible fibers f - λ for all but finitely many λ. The proof builds on Bergman's centralizer theorem and Cohn's stable-association theory, with Lemma 3.1 as the key technical step. Two applications are developed: Theorem 4.3 characterizes inclusion of eigenlevel sets of matrix evaluations by the right-multiple condition f a = a h with g = p(h), and Corollary 4.4 characterizes equality of eigenlevel sets by f a = a g. Theorem 5.4 classifies locally quasiconvex symmetric free polynomials as either polynomials -f that are sums of hermitian squares or univariate compositions of a convex quadratic form of linear terms.","tokens_in":11380,"tokens_out":12126,"duration_ms":121258,"significance":"Assuming the cited external theorems, Theorem 3.2 is a valuable structural result: it gives a clean centralizer criterion for compositeness and connects classical Bertini irreducibility with noncommutative factorization. The paper is generally careful and self-contained modulo standard results of Bergman and Cohn, and it introduces useful auxiliary facts such as the degree bound in Lemma 2.2. The eigenlevel and quasiconvexity applications are natural and likely to be useful in free real algebraic geometry. The main weakness is not in the central theorem, which appears sound, but in several application-side arguments that are too compressed and need explicit justification before the paper can be accepted.","major_comments":[{"comment":"The assertion that \\tilde f := f(y_1 + y_1^*, ..., y_d + y_d^*) is irreducible in C<y,y*> because f is irreducible over C is not justified. A factorization of \\tilde f could in principle become a trivial factorization of f under the substitution y_j = y_j^* = x_j/2, since the substituted factors might become constants or zero. Because Proposition 5.1 is load-bearing for Theorem 5.4, please provide a proof of this irreducibility statement or cite a result covering the complexification step.","section":"Proposition 5.1, proof"},{"comment":"After applying Theorem 3.2, the proof asserts f = p(h) with p ∈ R[t] and h ∈ R<x>, and then says that since f is symmetric, h is also symmetric because it is unique up to a scalar multiple. Two load-bearing points are passed over. First, Theorem 3.2 is stated over an algebraically closed field, so one must justify Galois descent from C to R for p and h. Second, uniqueness of h only yields h^* = c h with |c| = 1, and the antisymmetric possibility h^* = -h must be ruled out using the standing assumption that -f is not a sum of hermitian squares. Please replace this sentence with a lemma or detailed argument establishing the descent and the symmetrization.","section":"Theorem 5.4, proof (i)⇒(iii)"},{"comment":"The normalization step 'By comparing det(h_1(Ω^{(n)}) - λ_1 I), det(h_2(Ω^{(n)}) - λ_2 I) one can replace h_2 with αh_2 + β' is too compressed. Equality of eigenlevel hypersurfaces gives equality of the zero sets of the two determinants, not equality of the determinants themselves, and the affine normalization is then used to reach equation (4.2), which supplies the input for stable association via [HKV18, Theorem 4.3]. Please expand this reduction, including the justification of the determinant comparison for large n.","section":"Theorem 4.3, proof (i)⇒(ii)"}],"minor_comments":[{"comment":"The sentence 'Let /CZ be the algebraic closure of a field /CZ' uses the same symbol for the base field and its algebraic closure; please denote the base field and closure by different letters.","section":"Theorem 3.2, statement"},{"comment":"There is a typo in 'over a an algebraically closed field'; it should read 'over an algebraically closed field'.","section":"Introduction, page 1"},{"comment":"The statement begins 'Le f' and should read 'Let f'.","section":"Theorem 5.4, statement"},{"comment":"The equality h(Ω^{(n)}) = a(Ω^{(n)})^{-1} f(Ω^{(n)}) a(Ω^{(n)}) requires that a(Ω^{(n)}) be invertible for large n; this should be justified via Lemma 2.4 applied to the nonzero polynomial a rather than asserted.","section":"Theorem 4.3, proof (ii)⇒(i)"},{"comment":"The formula refers to a hypersurface in M_n(/CZ)^g but the dimension parameter should be d (the number of variables); please correct the notation.","section":"Corollary 4.1 and equation (4.1)"},{"comment":"The reference [HKMV] is cited without a year or arXiv number; please add complete publication data or mark it clearly as a preprint.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The central theorem is strong and the proof appears sound, so the paper is publishable in principle. The major comments are concentrated in the application sections and are repairable by adding detailed arguments rather than by changing the main results. If the author supplies the missing justifications, I would be happy to see the paper accepted."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper proves a true free-algebra analogue of Bertini: if f is a noncommutative polynomial and f-λ factors for infinitely many λ, then f is composite (a univariate composed with a noncommutative h). That equivalence - actually four equivalent conditions in Theorem 3.2 - is new and is exactly the kind of result that makes the free algebra factorization theory feel less piecemeal. The proof is also better than one might expect: Lemma 3.1 reduces the infinite family of factorizations to a stable-association finiteness statement, and the centralizer argument via Bergman and Cohn is careful. I checked the step where b2 b1^{-1} is shown to be a rational function of f; the determinant argument plus degree comparison does the job. I did not find a hole.\n\nThe applications are meaningful. The eigenlevel-set characterization (Theorem 4.3) gives an algebraic certificate for inclusion of eigenlevel sets, and the quasiconvexity classification (Theorem 5.4) is a clean dichotomy: either -f is a sum of hermitian squares, or f is a univariate composed with a convex quadratic. The paper also contains a nice example (Example 4.5) showing the degree bound in Lemma 2.2 is optimal, which I appreciate.\n\nSoft spots are minor and mostly expository. In Theorem 5.4, the claim that \"since f is symmetric, h is also symmetric because it is unique up to a scalar multiple\" is too terse. The uniqueness comes from Lemma 3.1, but the passage from C to R needs a sentence or two; as written it reads like an assertion rather than a proof. Similarly, several applications lean on [HKV18, Theorem 4.3] and [HKMV, Theorem 1.5]; those are legitimate external tools, but the paper doesn't spell out exactly how the constant-term perturbation is handled in Corollary 4.1. A referee should ask for these two points to be expanded, not for any mathematical change.\n\nThe citation pattern is fine. The author uses his own prior work with Helton, Klep, and McCullough, but as input to the applications, not to define the target conclusions. The core theorem stands on Bergman and Cohn.\n\nWho is this for: anyone working in free analysis, noncommutative factorization, or free real algebraic geometry. It deserves a serious referee. My own verdict would be accept after minor revision.","headline":"Genuine free-algebra Bertini theorem with a clean proof and solid applications; only minor exposition gaps.","tokens_in":11886,"tokens_out":2202,"would_cite":true,"duration_ms":21922,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["16U30","13P05","47A56","52A05"],"pacs":[],"model":"deepseek-v4-flash","headline":"A noncommutative polynomial whose shifts $f-\\lambda$ factor for infinitely many $\\lambda$ must be a composition $p(h)$ of a univariate polynomial $p$ and a noncommutative polynomial $h$.","keywords":["free irreducibility theorem","noncommutative polynomial","factorization","composition","eigenlevel set","quasiconvexity","free algebra","centralizer"],"falsifier":"Exhibit a nonconstant $f$ over an algebraically closed field and an infinite set of scalars $\\lambda$ such that $f-\\lambda$ factors in the free algebra for every $\\lambda$ in the set, yet $f$ is not a composition $p\\circ h$ with $\\deg p>1$; equivalently, find a non-composite $f$ whose centralizer in the universal skew field contains an element outside $\\mathbb{F}(f)$.","tokens_in":10986,"feed_emoji":"🧮","tokens_out":11833,"duration_ms":93406,"temperature":0.7,"pith_summary":"This paper proves a free-algebra analog of the classical irreducibility theorem of algebraic geometry: for a nonconstant noncommutative polynomial $f$, the shifted polynomials $f-\\lambda$ can factor nontrivially for infinitely many scalars $\\lambda$ only when $f$ is itself a composition $f=p\\circ h$ of an ordinary univariate polynomial $p$ and a noncommutative polynomial $h$. Equivalently, a non-composite $f$ has $f-\\lambda$ irreducible for all but finitely many $\\lambda$. The proof rests on the centralizer theorem for free associative algebras: when $f$ is not a composition, the elements commuting with $f$ in the universal skew field are exactly the rational functions of $f$. This theorem is then used to give algebraic certificates for when eigenlevel sets of matrix evaluations are nested or equal, and to classify locally quasiconvex noncommutative polynomials as either negative sums of hermitian squares or univariate compositions of a convex quadratic form.","feed_headline":"If f − λ factors infinitely often, f is composite","feed_subtitle":"Endlessly factorable noncommutative shifts collapse to a univariate composition, settling eigenlevel and convexity.","key_machinery":"The central object is the centralizer theorem for free associative algebras: for a non-composite $f$, the subalgebra of elements commuting with $f$ consists exactly of the polynomials in $f$, and in the universal skew field of fractions the centralizer is $\\mathbb{F}(f)$. Around this, the proof uses stable association, the notion that two polynomials occupy the same two-sided ideal up to invertible matrix factors in the free ideal ring, together with the finiteness of stable-associated classes and a degree-reduction lemma. Those ingredients force right factors of $f-\\lambda$ across different $\\lambda$ to be proportional. For the geometric applications, irreducibility results for free loci of noncommutative polynomials convert the algebraic irreducibility of $f-\\lambda$ into reducedness and irreducibility of eigenlevel hypersurfaces.","core_discovery":"Theorem 3.2 asserts that for an algebraically closed field and a nonconstant $f$ in the free algebra, four conditions coincide: (i) $f-\\lambda$ factors for infinitely many scalars $\\lambda$; (ii) $f-\\lambda$ factors for every $\\lambda$; (iii) the centralizer of $f$ is strictly larger than $\\mathbb{F}[f]$; and (iv) $f$ is composite over $\\mathbb{F}$, i.e. $f=p\\circ h$ with $\\deg p>1$ and $h$ noncommutative. The nontrivial direction uses a lemma showing that if $f b_1=b_1 g$ and $f b_2=\\alpha b_2 g$ with degrees below $\\deg f$, then $\\alpha=1$ and $b_2$ is a scalar multiple of $b_1$; the centralizer theorem provides the proportionality. Applying this to the finitely many stable-associated candidates for the right factors of $f-\\lambda$ gives a contradiction unless $f$ is composite. Consequently, non-composite $f$ have $f-\\lambda$ irreducible for all but finitely many $\\lambda$; with existing irreducibility results for free loci, their eigenlevel sets are reduced irreducible hypersurfaces for large matrix sizes. The same machinery yields the eigenlevel inclusion certificate and the locally quasiconvex classification.","pith_inferences":["Because stable association reduces to a finite linear system, the proof suggests a direct computational test for compositeness: search for degree-reducing witnesses by solving linear equations in the free algebra. The paper does not present such an algorithm.","The eigenlevel certificate $f a = a g$ is an intertwining relation; a natural extension is to ask whether a similar criterion governs inclusion of spectra of noncommutative rational or analytic free functions.","The locally quasiconvex classification may imply that every locally quasiconvex $f$ whose positivity domain is proper is LMI-representable; checking small examples in low degrees could reveal whether the classification survives beyond polynomials."],"forward_implications":["For every non-composite $f$, the family $\\{f-\\lambda\\}$ is irreducible for a cofinite set of $\\lambda$; factorization of shifts is a compositional phenomenon.","Eigenlevel inclusion is certified algebraically: if every eigenlevel set of $f$ lies in an eigenlevel set of $g$, then $g=p(h)$ and $f a = a h$ for some nonzero $a$; equality of eigenlevel sets holds exactly when $f a = a g$.","A homogeneous $f$ has $f-1$ factor in the free algebra if and only if $f=f_0^n$ for some $n>1$ and homogeneous $f_0$.","Locally quasiconvex symmetric polynomials are classified: either $-f$ is a sum of hermitian squares, or $f=p(\\ell_0+\\ell_1^2+\\cdots+\\ell_m^2)$ with explicit conditions on $p$; the convexity then holds for every $\\lambda>0$."],"supporting_citations":[{"why":"Supplies the centralizer theorem for free associative algebras, the key ingredient of Lemma 3.1.","marker":"[Ber69]"},{"why":"Provides the free ideal ring theory of stable association, unique factorization, and the centralizer description in the universal skew field, along with the finiteness result used in Theorem 3.2.","marker":"[Coh06]"},{"why":"Theorem 4.3 on reducedness and irreducibility of free loci converts algebraic irreducibility of $f-\\lambda$ into the eigenlevel geometry of Corollary 4.1.","marker":"[HKV18]"},{"why":"Theorem 1.5 on irreducible symmetric polynomials with convex positivity domains drives Proposition 5.1 and the quasiconvex classification.","marker":"[HKMV]"},{"why":"Establishes that convex free semialgebraic sets have LMI representations, supplying the convexity criterion behind Theorem 5.4.","marker":"[HM12]"},{"why":"Defines quasiconvex free polynomials and proves the base classification that local quasiconvexity extends.","marker":"[BM14]"},{"why":"Positive noncommutative polynomials are sums of hermitian squares, pinning down the everywhere-negative-semidefinite case.","marker":"[Hel02]"}],"fun_headline_variants":["Composite free polynomials from infinite factorability","Infinite factorability forces free polynomial composition","Free Bertini: infinitely factorable means composite","Eigenlevel sets pinned by free composite structure","When f−λ factors infinitely, f is composite"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof leans on the centralizer theorem: for a non-composite $f$, everything in the universal skew field that commutes with $f$ is a rational function of $f$; if that statement failed, the proportionality argument would break and Theorem 3.2 would lose its proof.","fun_headline_variants_meta":{"raw":{"variants":["Composite free polynomials from infinite factorability","Infinite factorability forces free polynomial composition","Free Bertini: infinitely factorable means composite","Eigenlevel sets pinned by free composite structure","When f−λ factors infinitely, f is composite"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000212,"raw_usage":{"total_tokens":1481,"prompt_tokens":1072,"completion_tokens":409,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":688,"completion_tokens_details":{"reasoning_tokens":340}},"tokens_in":688,"tokens_out":409,"duration_ms":4523,"temperature":1.0,"reasoning_tokens":340,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:27:06.142674+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Exhibit a nonconstant $f$ over an algebraically closed field and an infinite set of scalars $\\lambda$ such that $f-\\lambda$ factors in the free algebra for every $\\lambda$ in the set, yet $f$ is not a composition $p\\circ h$ with $\\deg p>1$; equivalently, find a non-composite $f$ whose centralizer in the universal skew field contains an element outside $\\mathbb{F}(f)$.","supporting_citations":[],"review_version":1}